Question

Consider the line integral $I={\int }_c(x^2+i{y}^2)dz$, where $z=x+iy$. The line $c$ is shown in the figure below



The value of $I$ is

• A. $\frac{1}{2}i$
• B. $\frac{2}{3}i$
• C. $\frac{3}{4}i$
• D. $\frac{4}{5}i$

Answer 1

The correct option is B $\frac{2}{3}i$

Given integral $I={\int }_c(x^2+i{y}^2)dz,z=x+iy$ the line $C$ is a straight line passing through origin and its equation is givne by $y=x$
Substittuing $y=x$ in given integral we get
$I={\int }_c(x^2+i{x}^2)(dx+idx)$
$=I={\int }_c{x}^2(1+i)(1+i)dx$



$={\int }_c{x}^2(1-1+2i)dx$
$={\int }_{c}2i{x}^{2}dx=2i{\int }_0^1{x}^{2}dx$
$=2i{\left[\frac{x^3}{3}\right]}_0^1=2i[\frac{1}{3}-\frac{0}{3}]$
$=\frac{2i}{3}$